Heavy snowfalls and icy weather conditions are likely to continue throughout the week, bringing further chaos to the roads and public transport system, the Met Office said today.

Today's disruptions to trains, tubes and buses, which resulted in one in five people not making it to work, will probably persist until Friday, although the weather will become less severe as the week progresses.

Significant snowfalls are likely in some places. Icy conditions are set to prevail in the south-east this evening, and sleet and more snow are likely further north. The Met Office has issued a severe weather warning.

As much as 20-30cm of snow is possible on the higher ground, with drifting occurring in fresh easterly winds.

Another band of heavy snow is moving across from France and could drop 10-20cm on Kent, Surrey and eastern parts.

A Met Office spokesperson, Sarah Holland, said: "Temperatures are going to barely reach freezing today because of the cold easterly winds moving in from northern Europe. There will be more snow coming through again tomorrow, though lower levels might be a bit slushy.

"The rest of the week is going to remain cold. There could still be snow on Thursday. Because we have been used to milder winters, for many this snow is a surprise."

Those travelling to work in London today were hit hard after the heaviest snowfall in decades disrupted almost all forms of public transport. The Met Office issued an extreme weather warning for London and the south-east of England, with more than 15cm of snow falling in London and 25cm on the North Downs.

The snow arrived on a freezing easterly wind after gathering strength over the North Sea. Driving conditions are treacherous and Scotland Yard said it had received reports of a number of crashes in the early hours of today, though no one was seriously hurt.

All London bus services were withdrawn because of fears for passenger safety, leaving the snow-laden city streets unusually quiet. A limited bus service resumed this afternoon.

The congestion charge was suspended in London for one day as the capital was faced with "some of the most challenging weather conditions" seen in the city for two decades, the mayor, Boris Johnson, said.

Rural roads and some main roads in Sussex were described as treacherous by police as they urged motorists to consider whether they really had to drive today. Police said several bands of heavy overnight snowfall had affected mainly East Sussex and mid-Sussex. The A27 in the Hollingbury and Falmer areas of Brighton was impassable after seven articulated lorries became stranded in the drift. The longest queue of traffic stretched 32 miles on the M25 motorway between Watford and Reigate.

Southeastern said it had cancelled all rail services into London after heavy snow fell overnight in Kent and Sussex. A spokeswoman said: "We are endeavouring to run coastal services in Kent and East Sussex but all trains to London are cancelled."

Airports were also badly hit as BAA warned passengers travelling to and from Heathrow and Stansted airports to expect "significant delays and cancellations". Passengers were advised to check with their airline before leaving for the airport. The London Ambulance Service said it would only respond to "life-threatening" emergency calls because it was under severe pressure and had received more than 650 calls between midnight and 7am today.

Richard Webber, acting director of operations, said: "The extreme weather has placed the London Ambulance Service under severe pressure today and our priority is to get to those patients with life-threatening illnesses or injuries as quickly as possible. Due to current levels of demand we will, until further notice, only be sending ambulances to patients across the capital with life-threatening illnesses or injuries. We would urge Londoners to use their ambulance service wisely and only call 999 in the event of a life-threatening emergency."

The AA has warned motorists to ensure they keep warm clothes in the car after dealing with unprepared drivers "flirting with hypothermia" at breakdowns during the last big freeze.

A police spokesman said: "Police advice to drivers is to consider whether they need to drive at all. If they have to go out, check national and local weather forecasts."

Drivers were advised to be prepared for the conditions if they needed to travel and ensure they go with boots, a spade, warm clothes and food and drink.

Many businesses were also expected to close down for the day as workers struggle to make journeys on icy roads.

A spokesman for the Highways Agency said the A66 in the north-west of England, between Brough and Stainmore Summit, was closed at 1.35am after a request from the police. He said it was likely it would remain closed all day. Two lanes of the M62 over the Pennines were also closed for several hours last night but have now reopened.

Many schools in the south-east were forced to close due to the weather. More than 50 schools were closed in Kent, and Westminster council announced all its schools would remain closed. Schools in Birmingham and the Black Country have also closed their doors.

The Department for Children, Schools and Families said it expected to get a clearer picture of how many schools were closed nationally later in the day. A spokesman said it was hoped schools would remain open where possible.

"It is for individual schools to make any decision to close but we expect headteachers to take a common sense approach based on the conditions at the school and also the safety of any pupils travelling," he said. "We will be monitoring the situation day by day and we would hope that headteachers can keep schools open while it is safe for them to do so, but decisions about closure have to be taken locally by those who know the local conditions."

The Conservative leader, David Cameron, had to cancel a visit to Haberdashers' Aske's Hatcham college, an academy in New Cross, south-east London, where he was due to highlight a package of proposals aimed at improving numeracy, including a maths taskforce headed by the former Countdown host, Carol Vorderman.